On the girth of extremal graphs without shortest cycles Original Research Article

Let image denote the set of graphs image of order image that contain no cycles of length less than or equal to image which have maximum number of edges. In this paper we consider a problem posed by several authors: does image contain an image cycle? We prove that the diameter of image is at most image, and present several results concerning the above question: the girth of image is image if (i) image, diameter equal to image and minimum degree at least 3; (ii) image, image and image. Moreover, if image we find an extremal graph of girth 8 obtained from a 3-regular complete bipartite graph subdividing its edges. (iii) We prove that if image and image the girth is at most image. We also show that the answer to the question is negative for image.